Imaging heart motion using harmonic phase MRI - …iacl.ece.jhu.edu/pubs/p111j-ieee.pdfImaging Heart...

186 IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. 19, NO. 3, MARCH 2000

Imaging Heart Motion Using Harmonic Phase MRINael F. Osman, Elliot R. McVeigh, and Jerry L. Prince*

Abstract—This paper describes a new image processingtechnique for rapid analysis and visualization of tagged cardiacmagnetic resonance (MR) images. The method is based on theuse of isolated spectral peaks in spatial modulation of magne-tization (SPAMM)-tagged magnetic resonance images. We callthe calculated angle of the complex image corresponding to oneof these peaks a harmonic phase (HARP) image and show thatHARP images can be used to synthesize conventional tag lines,reconstruct displacement fields for small motions, and calculatetwo-dimensional (2-D) strain. The performance of this newapproach is demonstrated using both real and simulated taggedMR images. Potential for use of HARP images in fast imagingtechniques and three-dimensional (3-D) analyses are discussed.

Index Terms—Image processing, motion estimation, MR tag-ging, MRI.

I. INTRODUCTION

OVER the last decade, cardiac imaging using tagged mag-netic resonance (MR) imaging has become an established

technique in medical imaging [1]–[4]. MR tagging uses a specialpulse sequence to spatially modulate the longitudinal magneti-zation of the subject prior to acquiring image data. Over manyheartbeats acquired in a single breath hold, enough data can beacquired to reconstruct an image sequence in which the tag pat-tern is deformed by the underlying motion of the heart [5]. Usu-ally, the tagging process is thought of as producing saturatedplanes orthogonal to the image plane, leading to images such asthat shown in Fig. 1(a). It is not necessary to fully saturate thesignal, however, and two-dimensional (2-D) sinusoidal patternssuch as that depicted in Fig. 1(b) are also possible. The patternsdepicted in Fig. 1(a) and (b) are called spatial modulation ofmagnetization (SPAMM) tag patterns [6], [7]. This class of pat-terns forms the basis for the work presented in this paper.

Although significant improvements in the MR tagged imageacquisition methodology has occurred [6], [8]–[12], lack of fastquantitative analysis and visualization techniques is preventingMR tagging from being widely adopted in the clinical setting.Most analysis techniques in MR tagging have used image

Manuscript received November 9, 1998; revised December 16, 1999. Thisresearch was supported by the National Heart, Lung, and Blood Institute underGrant R01-HL47405 and in part by the National Science Foundation underGrant MIP-9350336. The Associate Editor responsible for coordinating the re-view of this paper and recommending its publication was J. Duncan.Asteriskindicates corresponding author.

N. F. Osman is with the Center for Imaging Science, Department of Elec-trical and Computer Engineering, The Johns Hopkins University, Baltimore,MD 21218 USA.

E. R. McVeigh is with the Department of Biomedical Engineering, The JohnsHopkins University, Baltimore, MD 21218 USA.

*J. L. Prince is with the Department of Biomedical Engineering, The JohnsHopkins University, Baltimore, MD 21218 USA (e-mail: [email protected]). Heis also with the Center for Imaging Science, Department of Electrical and Com-puter Engineering, The Johns Hopkins University, Baltimore, MD 21218 USA.

Publisher Item Identifier S 0278-0062(00)02981-5.

processing techniques to detect tag features which are thencombined into a detailed motion map (displacement and/orstrain) using interpolation. To detect tags, for example, Guttmanet al.[13] used morphological image processing and matchedfiltering techniques [13], Young and Axel [1] and Kumar andGoldgof [14] used deformable meshes, and Young and Axelhave used manually identified points [1]. To interpolate a densemotion, Young and Axel [1] used a finite element model, O'Dellet al.[15] used a truncated polynomial expansion, Denney andPrince [16] used a stochastic estimation scheme, and Radevaetal.[17] used a three-dimensional (3-D) B-spline.

There are several disadvantages to the existing analysismethods. First, manual intervention is almost always requiredin feature detection [1], [13]. Although progress is being madein further automating this step (cf. [18]), it is not clear thatthese methods will ever be completely automatic. Second,because features must by necessity be distinct, interpolationwill always be required to achieve dense motion estimation.Third, since the endocardial and epicardial boundaries of theleft ventricle are most often explicitly used in the interpolationprocess, modification to estimate right ventricular motionwill require different software and new modeling approaches.Finally, the combined requirements of manual intervention andinterpolation makes these methods very time consuming. Onecannot hold out great promise that these methods will ever beviable in near real-time diagnosis. The approach described inthis paper addresses every one of these concerns.

Optical flow methods have also been explored in the analysisof tagged MR image sequences [19]–[21]. In this approach, si-nusoidal tag patterns are used instead of saturated planes. Imagebrightness gradients become features and together with tem-poral derivatives estimated from image pairs they can be used toproduce dense motion estimates. Generally, regularization is re-quired in order to account for the fact that brightness gradientscontain information about motion only in the direction of thegradient. It is possible, however, to combine information frommultiple directions, similar in spirit to what is done in planartagging, in order to produce optical flow estimates without reg-ularization [21]. One limitation of this overall approach is that itis difficult to measure large motions such as that occurring be-tween end diastole and end systole. Instead, successive motionsbetween image frames are measured and tracking is required tomeasure the total motion [19].

Other techniques exist to measure heart motion using MRwithout tagging. Velocity encoding techniques use specializedpulse sequences to produce a complex image in which thephase is linearly dependent on the motion [22]. These methodsare useful in measuring the flow of body fluids such as bloodand cerebrospinal fluids, but cannot measure the large motionspresent in the heart. Instead, tracking (exactly as in optical

0278–0062/00$10.00 © 2000 IEEE

OSMAN et al.: IMAGING HEART MOTION USING HARMONIC PHASE MRI 187

Fig. 1. (a) A 1-D SPAMM-tagged image. (b) A 2-D SPAMM-tagged image. Each shows a short-axis view of the LV of the heart. (c) and (d) are the magnitudes ofthe Fourier transforms of (a) and (b), respectively. The elongation of the peaks in (d) is orthogonal to the elongated geometry of the cross section of the body in (b).

flow) is required to measure gross motion from successivetemporal measurements [23]–[25]. It is also possible to directlymeasure the temporal derivative of strain, the so-called strainrate [26], [27], but this method is also limited to small motions.Other variations on velocity encoding exist, including thesimultaneous use of velocity encoding and tagging [24], [28].The primary limitations of all the velocity encoding approachesare the inability to measure large motions, low noise immunity,and susceptibility to motion artifacts.

In this paper, we describe a new approach to analyze car-diac tagged MR images using the concept of harmonic phase(HARP) images. The method is based on the fact that SPAMM-tagged MR images [6], [7] have a collection of distinct spectralpeaks in the Fourier domain, and that each spectral peak con-tains information about the motion in a certain direction [11].The inverse Fourier transform of just one of these peaks, ex-tracted using a bandpass filter, is a complex image whose phaseis linearly related to a directional component of the true motion.We define a HARP image to be the principal value of the phase

of this complex image, a number that is constrained to lie in therange by the wrapping action of the standard inversearctangent operator. We show in this paper that HARP imagescan be used to: estimate synthetic tag lines; measure small dis-placement fields; and compute 2-D strain.

The motion computation methods we describe in this paperuse image data corresponding to a single time instant in the car-diac cycle. They do not require a sequence of images as in someother techniques. This distinguishes the methods we describeherein from those in our recent paper describing a technique totrack points through images sequences using HARP concepts[29]. The calculations we describe are automatic, fast, and ex-tendable to three dimensions. We show initial experimental re-sults that show promise for fast fully automated imaging of car-diac strain. Several areas for extensions and improvements, in-cluding an approach for rapid imaging tuned to the HARP pro-cessing methodology, are also described.

The use of local phase information in the estimation of mo-tion from images is not new to HARP. This concept was in fact


developed by several researchers approximately a decade ago[30]–[34] and is now widely known as phase-based optical flow[35], [36]. HARP, however, is different in several respects fromthe conventional phase-based approaches found in the literature.First, the methods described in this paper are based on the anal-ysis of a single image rather than an image sequence. Thus theconcept of velocity-matched, spatio-temporal filters (cf. [37])cannot be used because an image sequence is not presumed to beavailable. Second, because of the physical MR tagging process,only two spatial filters are required to unambiguously compute2-D motion and this motion is related to the true 3-D motionin a known way. These filters are designed according to the ex-pected cardiac strain rather than using velocity-tuned filters asin [32]. Furthermore, Gabor filters are not appropriate in ourapplication because they would unnecessarily attenuate spec-tral information that should contribute to the motion computa-tions. Finally, in addition to our computation of small displace-ments, which is similar to the phase-based optical flow methodsof Fleet and Jepson [32], HARP uses local phase information tocompute local strain. This is an important contribution to themotion-from-phase literature.

II. SPAMM-TAGGED IMAGES

Fig.1(a) and (b) shows two SPAMM-tagged MR images, eachshowing a short-axis view of a human left ventricle (LV). Thespectral peaks appearing in the magnitude of the Fourier trans-forms of these images, shown in Fig. 1(c) and (d), arise be-cause the tagging process modulates the underlying image witha SPAMM tag pattern. In this section, we develop a mathemat-ical model describing the effect of heart motion on SPAMM-tagged images. In subsequent sections, we present methods toestimate motion from the acquired images.

Our focus throughout this paper is on 2-D MR SPAMM-tagged images because they can be readily acquired on any MRscanner while 3-D image sets are more difficult to prescribe andacquire. Since the heart motion is 3-D, however, we are carefulto precisely relate the 2-D quantities we estimate to the true3-D motion. In this way, the limitations of our methods can beclearly understood and approaches to extend these methods to3-D imaging can be clearly outlined.

A. SPAMM Tagging

Tagging pulse sequences are usually imposed at end diastole,a time in the cardiac cycle when the left ventricle is full of bloodand the heart is relatively slow-moving. The QRS complex ofthe ECG signals the moment of end diastole. Present technologyrequires multiple heartbeats, assumed to be perfectly repeating,at least from end diastole to end systole. Under this assumption,the imaging equations governing MR tagging can assume thatthe entire imaging process takes place in one heart beat. Further-more, end diastole can be considered to be time and theposition of points within the heart at end diastole can be treatedas a material coordinate system, denoted by points . Letus now consider the effect of the tagging process itself, assumedto take place at with a stationary heart.

The tagging process imposes a temporary spatial variation inthe longitudinal magnetization of the protons inside the body

[7], [8], [38]. SPAMM tag patterns are imposed by applyinga sequence of hard radio frequency (RF) pulses, generallywith different tip angles, with seconds between each pulse.A gradient waveform is applied between these pulses andthe whole sequence is followed by a crusher to remove the effectof transverse magnetization [7]. The simplest pulse sequencehas only two RF pulses and it is called a1-1 SPAMMpulsesequence. Now suppose that an image passing through pointiscreated immediately after application of the SPAMM tag pulsesequence. Its value is given to good approximation by

(1)

where is the value that would have been imaged withouttagging and the tag pattern is given by (cf. [39])

(2)

where are coefficients determined by the sequence of tip an-gles and the gradient directionis given by ,where is the gyromagnetic ratio. The tag pattern in (2) isone-dimensional (1-D) because its values are identical in planesorthogonal to . A 2-D SPAMM tag pattern is generated by ap-plying two 1-D SPAMM pulse sequences in rapid successionusing two different gradient directions and . The resultingimage value at point is

(3)

where it is assumed that the protocols of the two 1-D patternsare identical except for the gradient direction.

Both (1) and (3) describe amplitude modulation of theunderlying signal intensity by a pattern of cosines. Because acosine has two symmetric spectral peaks in Fourier space, a1-D SPAMM pattern generated with RF pulses hasspectral peaks. An example of such a pattern is shown inFig. 1(a). Because a 2-D SPAMM pattern is the product oftwo 1-D SPAMM patterns, it has spectral peaks inthe Fourier domain. An example of a 2-D SPAMM pattern isshown in Fig. 1(b). The locations of the spectral peaks inFourier space are easily found fromin the 1-D case or and

in the 2-D case and the resulting tagged image intensity canbe written in the following unified manner:

(4)

where for 1-D SPAMM and for2-D SPAMM. The coefficients are readily determined from

.

B. Motion and Images

In order to relate calculations made using 2-D tagged imagesto the actual 3-D motion, it is necessary to establish a mathe-matical relationship between the motion of the heart, the phys-ical position of pixels within an image, and the intensities oftagged images.


As the heart deforms, a material point within the myocardiummoves from its reference positionto a new spatial positionat time . The reference map characterizes this motionby giving the material position of the spatial point at time .Using in place of in (4) gives a relationship involvingthe spatial coordinate instead of the material coordinate.

Now suppose the location of a pixel within an image is desig-nated by the 2-D image coordinate vector . Thenthe 3-D position of the pixel at image coordinate can bedescribed by the function where

and are two 3-D orthogonal unit vectors describing theimage orientation and is an image origin. This function canbe written in matrix notion as follows:

(5)

where . Using in and substitutingfor in (4) yields the following expression for a

SPAMM-tagged MR image:

(6)

where

(7)

Here, the coefficients are made functions of time to accountfor the fading of tag patterns caused by longitudinal relaxationand the imaging pulse sequence. Equation (6) shows that aSPAMM-tagged image is the sum of complex images, whichwe call harmonic images, each corresponding to a distinctspectral peak identified by the frequency vector.

In the following development we implicitly assume that allgradient directions are oriented parallel to the image plane.That is, they are linear combinations of and . This impliesthat are also parallel to the image plane. Thisis the tagging protocol most commonly implemented in practiceand is best for imaging the desired 2-D motion quantities froma single image plane. The consequences of usingpointing outof the plane (cf. [40]) are interesting since there is the potentialto directly estimate certain 3-D motion quantities but such adevelopment is beyond the scope of the present paper.

III. H ARMONIC PHASE IMAGES

A. Displacement Modulation

Interpretation of the action of motion as a modulation processis key to the methods developed herein (see also [41], [42]). Wecan see this relationship by examining theth harmonic image

that corresponds to theth spectral peak in the Fouriertransform of . One important description of motion is thedisplacement field, defined as

(8)

Using this expression in (7) yields

(9)

where . Replacing by theexpression in (5) and expanding yields

(10)

The first exponential term in (10) represents a simple complexsinusoidal carrier with frequency and phase . Thiscarrier determines the position of the spectral peak at inthe Fourier domain. The term multiplies this complexsinusoid, so it represents amplitude modulation (AM) in analogywith communications theory. Using the same analogy, the lastterm in (10) is the most interesting term, as it represents a phasemodulation (PM) of the underlying carrier by the displacementfield . This property is what we exploit in our development ofmotion estimation methods. To complete the analogy with com-munications theory, it has been in shown in [41] that the gradientof the phase, which we called the local spatial frequency is mod-ulated by the local strain. This is analogous to frequency mod-ulation (FM) in communications theory.

Although the spectrum of is spread throughout the Fourierdomain, most of its energy is concentrated, due to the nature ofthe LV motion, around the spectral peak located at . Theextent of the energy localization is affected by the motion, whichin the case of contracting heart is manifested by the spreadingof the spectral peaks. It is worth mentioning that a rigid rotationof the pattern would change the angular position of the spec-tral peaks. Fortunately, the heart's twisting during systole causesonly a small angular motion of the spectral peaks. It is possibleto extract an estimate of the harmonic image using a 2-D band-pass filter centered around whose size is large enoughto capture the spectral peak after maximum spreading. This es-timate will differ from the truth because of both sampling andnoise effects and also because of both energy lost outside of thebandpass region and energy entering the bandpass region fromother spectral peaks. Designing an appropriate bandpass filteris largely a matter of trading off the loss of accuracy when thebandpass region is too small with the introduction of artifactswhen the bandpass region is too big. We give an empirical studyfor optimizing these filters in Section I-C. For now, we assumethat a bandpass filter is capable of extractingexactly.

B. HARP Images

Since the harmonic image is complex, it has both amagnitude and phase at each. From (7) we see that the phaseof is an image given by

(11)

which we refer to as aharmonic phase image. We define aHARPimageto be the calculated phase of, which is given by

(12)

where

otherwise(13)


(a) (b)

Fig. 2. (a) 2-D 1-1 SPAMM-tagged short axis image of the left ventricle and (b) one of its angle images. Notice that the branching and truncation in (b) occursoutside the myocardium in regions where the underlying image intensity in (a) is very small and noise dominates.

Because of the inverse tangent operator, a HARP image is theprincipal value of its corresponding phase image, and is re-stricted to be in the range . Formally, a HARP imageis related to the phase image as follows:

(14)

where the nonlinear wrapping function is given by

(15)

From (11), we see that is linearly related to the referencemap , so it could be used in a fairly straightforward manner tocalculate motion and motion-related quantities. Unlesshap-pens to fall into the range , however, to actually cal-culate requires the use of 2-D phase unwrapping techniques[43], which are very sensitive to noise. Although robust leastsquares techniques have been developed [44]–[46], these algo-rithms are fairly time-consuming and do not always work prop-erly [46]. We can expect this process to be particularly prob-lematic in the high-noise environment of cardiac tagged MRimaging. Instead, the motion estimation procedures describedin this paper rely only on having the HARP images, which arereadily computed in a very robust manner using (13). It shouldbe noted that both the phase and the HARP angle are materialproperties of the tagged tissue, so that if one follows a partic-ular material point , its phase and HARP angle remain constantwith time. This is an important property, which will be exploitedin Section III-C, where we describe the use of HARP images inmeasuring motion.

As a demonstration, we calculated HARP images for theimage sequence whose first image appears in Fig. 1(b) usingthe spectral peak circled in Fig. 1(d). A close-up image of the

LV near end systole is shown in Fig. 2(a) and its HARP imageis shown in Fig. 2(b). The strongest apparent features in thisHARP image are the lines of discontinuity traveling at a 45angle. These lines correspond to the transition in angle from

to caused by the wrapping in (15). Since these linescorrespond well to the apparent movement of the tag patternitself, they demonstrate the fact that the tag pattern phase is amaterial tissue property, remaining constant despite fading andintensity variations of the tag pattern itself.

C. Measuring Motion Using HARP Images

We now outline three ways to use HARP images in cardiacmotion analysis from tagged MR images. Experiments showingtheir use and performance are given in Section IV.

1) Synthetic Tag Lines:The discontinuities in Fig. 2(b)strongly resemble 1-D tag lines. In fact, the lines we see in thisfigure represent a crude (pixelated) approximation of imageisocontours having the value. Since the HARP angle of atagged image is a material property of the tissue, these linesrepresent samples of surfaces all having the angle. Thesesurfaces are completely analogous to tag surfaces arising fromplanar tagging or higher-order 1-D SPAMM tagging and thelines are analogous to the intersection of image planes withthese tag surfaces. In essence, HARP images can be used togenerate synthetic tag lines by identifying isocontours withinthe HARP images.

Mathematically, a set of tag lines is defined by

(16)

where is an arbitrary angle. Computationally,is determined by running an isocontour algorithm at


level on image . After restricting these lines to thosefalling within the LV myocardium, they can then be used in anystandard analysis procedure based on planar tags (cf. [6], [15]).

There are several advantages in using HARP images togenerate synthetic tag lines. First, it is a completely automaticprocess. Second, the tag lines will have subpixel resolutionsince good isocontour algorithms have this property (see resultsin Section IV). Third, the entire image will have these tagsautomatically identified, including those passing through bothleft and right ventricular myocardium. Finally, tags spaced veryclosely together can be generated by selectingvalues in therange so that tag lines will be synthesized withinover the spatial period . In principle, there is nofundamental limit on how close these tag lines can be spaced,because they are not limited by the detectability of featuresspaced close together.

2) Measuring Small Motions:Consider the point in theimage plane at time. The position of this material point atis given by the reference map , which is a point notusually in the image plane. Now suppose we measure two har-monic images of the form (10) having linearly-independent tagfrequencies and parallel to the image plane. Since thephases and of the harmonic images are material proper-ties, the point has the same phases as doesat .Furthermore, since and are parallel to the image plane,the orthogonal projection of onto the image planealso has these same phases. Although the true motion of anypoint in the heart is 3-D, the apparent motion in 2-D is uniquelydefined between the projection of the point on theimage plane and its spatial position. For small motions (to bedefined precisely below), we can directly measure the apparentmotion.

Given the complete knowledge of the tagging pulse sequenceand given the measured HARP image , the followingquantity can be calculated:

(17)

The 3-D displacement can be written aswhere . It is shown in Appendix A that,

provided that and are linearly independent and, the 2-D displacement within the image plane

can be calculated according to

(18)

where . This is our estimate of 2-D displacementfor small motions.

There are several ways to assure that the conditionsare satisfied to make (18) valid. First, it is possible

to image very shortly after end diastole, before there is substan-tial heart motion. This will be useful, and potentially clinicallyimportant in the first few tens of milliseconds of systole, butwill not allow the calculation of displacements throughout thesystolic phase. Second, if low-frequency tag patterns are used,the physical period of the tag pattern is larger, so larger mo-tions will not produce angle ambiguity (wrapping). The diffi-culty here is that the spectral peaks of low-frequency patterns

will often interfere with one another, leading to motion artifacts.A third possibility is to apply the tag pattern at a fixed offsetfrom end diastole and image shortly thereafter.

We note that (18) is a direct consequence of having two lin-early independent brightness constraint equations arising fromthe HARP images and . This equation represents a spe-cial case of the phased-based optical flow method proposed byJepson and Fleet [32]. Here, however, only two filters are suffi-cient to reconstruct an arbitrary 2-D displacement field becausethere is sufficient energy and isolation in the spectral peaksarising from the MR tagging process. The following sectionpresents an approach to the measurement of local strain, repre-senting a significant extension to the past literature on the use oflocal phase information. In particular, strain is estimated usingonly a pair of HARP images acquired at any time during thecardiac cycle (provided that the tags persist). The fact that noreference images are required and that the calculation is exactdespite large deformations is important to the practice of cardiacstrain imaging.

3) Measuring Strain:Assume we have measured twoHARP images having linearly independent frequenciesand

parallel to the plane. Let be the 2-Dreference map describing the point in the image plane havingthe same two phases at time as does the point at time. Now let be a unit vector in the image plane. The

apparent strain in the directionis given by

(19)

where is the tensor corresponding to the derivative ofwith respect to , represented in matrix form by

. One possible way to calculate this quantity wouldbe to first calculate using (18), which determines ,and then use (19). This calculation would be limited to smallmotions, however, and, as we now show, it is possible to calcu-late without this limitation.

It is shown in Appendix B that

(20)

where . From (14), we see that the gradient ofa HARP image is the same as the gradient of a phase imageexcept at the points of discontinuity caused by the wrappingoperation. At these points, the gradient is theoretically infiniteand practically very large. It is possible to shift the locationsof the wrapping artifact, however, by simply addingto theHARP image and rewrapping. Gradients calculated at locationsof the previous discontinuities are now equal to the gradient ofthe phase image at that location. Therefore

(21)

for where the modified gradient operator is definedby

otherwise(22)

Now, using (21) in (20) yields

(23)


Fig. 3. The elliptical passband of filterF .

where . Substituting (23) into (19) yields

(24)

which is our estimate of apparent strain in a 2-D image plane. Itcan always be computed provided that and are linearlyindependent, in particular, it is not limited by the small motionassumption.

IV. EXPERIMENTAL RESULTS

In this section, we present experimental results related to theacquisition of and uses for HARP images. The first set of experi-ments is designed to determine the best position, size, and shapeof the bandpass filters required to compute HARP images fromacquired SPAMM-tagged MR images. The second set of exper-iments is designed to demonstrate and quantify the performanceof the three applications of HARP images to motion estimation.

A. Computing HARP Images

1) Bandpass Filter:The first step in computing a HARPimage is to acquire its harmonic image , which can beestimated by applying a bandpass filter to the image . Thetrue Fourier spectrum of the harmonic imageis determinedby the frequency , the spectrum of the underlying image, andthe motion. Since its spectrum is strongly peaked around(see Section III-A), we positioned the bandpass region of thefilter at , as shown in Fig. 3. We chose an ellipse for theshape of the bandpass region because it has a simple geometrythat adequately captures the gross shape of most spectral peaks.We fixed the major axis of the ellipse to be in the direction of

because we expect that most of the spectral changes dueto motion will be in this direction.

Letting denote 2-D frequency, the bandpass filter weused is given in the Fourier domain by

(25)

where

(26)

Here, , where and are the major andminor radii of the elliptic bandpass region in units of rad/cm;is an Euler matrix corresponding to a rotation by ; and

(unitless) in all our experiments. The function

yields the desired position, size, and shape of the main ellipticalbandpass region. The actual filter is unity within this re-gion, but to reduce ringing artifacts it tails off in a Gaussianfashion outside this region.

Given the described bandpass filter, the harmonic image isestimated using where theoperators and represent the Fourier transform and itsinverse, respectively. The corresponding HARP image is thenestimated as (Section III-B). The radii ofthe ellipse, and , should be selected to minimize the errorin this estimate, since this affects all other estimated motionquantities. We now describe how to characterize this error.

Suppose it was possible to estimate the phasesand fromtwo estimated harmonic images (for example by phase unwrap-ping their corresponding HARP images). Then 2-D referencemap could be estimated as follows [see (32)]:

(27)

where . Using (32) and (27), the error in thisestimate can be written as

(28)

If the error were small, i.e., —thenand

(29)

In the simulation experiments described below, the error wassmall enough to use (29). Moreover, in the simulation, we knowthe true reference mapand we can compute using (11)and (14). Therefore, the error as can be computed ateach point in the image.

2) Parameter Optimization:Quantitative experimentsdesigned to optimize the selection of and for a giventag frequency were conducted. We used images generatedby a simulation program designed in our laboratory, an earlyversion of which was reported in [47]. The program simulatesthe normal deformation of the LV during systole and producesrealistic tagged MR images. The smooth motion field producedby this simulation might not capture some of the more abruptmotion patterns of normal myocardium, but should apply in thecase of diseased myocardium since it is less active.

A typical image generated from our simulator is shown inFig. 4(a), which can be compared to an actual image shownin Fig. 2(a). The image in Fig. 4(a) was simulated using a 2-DSPAMM tag pattern. The gradients between the RF pulses pro-duces a 6-mm tag period in the two diagonal directions. For eachdirection, the SPAMM pulse sequence has two RF pulses with a45 tip angle each. The simulated image represents an end-sys-tolic image, so it includes the effect of motion, producing themaximum spreading of the spectral peaks, and tag fading as-suming myocardial tissue parameters of ms and

ms. The image matrix size is 6464 with a 1.09-mmpixel separation. The nine spectral peaks present in this 2-D 1-1SPAMM pattern are evident in the magnitude of the Fouriertransform of this image, as shown in Fig. 4(b). Seven similarshort-axis images were produced simulating the appearance ofthe LV at end systole given fundamental 1-1 2-D SPAMM tag


(a) (b)

Fig. 4. (a) A simulated 2-D 1-1 SPAMM-tagged image with a 6-mm tag period. (b) The magnitude of its Fourier transform.

periods of 2, 3, 4, 5, 6, 7, and 8 mm. We also obtained the true3-D reference map, so we could calculate the root-mean-square(rms) displacement error over the simulated myocardium.

Two bandpass filters and were used to estimate theHARP images and . The center frequencies of these filterswere given by

and

where is the tag period. To determine the best radii for thefilters, the rms error was computed for different values ofthe radii and . Those radii yielding the lowest rms errorfor each tag period, the optimal radii, are plotted in Fig. 5(a).There are two reasons why the optimal radii increase with in-creasing tag frequency. First, it can be shown theoretically thatthe amount of frequency spreading caused by the phase modu-lation effect of motion increases with increasing tag frequency.This is a standard result in communications theory. Second, asthe spectral peaks get farther away from the origin, there isless interference with the spectral peak at dc. The optimal radiiplotted in Fig. 5(a) were used in all subsequent experiments inthis section for both real and simulated data.

Fig. 5(a) also shows the rms error of the optimal radii plottedas a function of the tag period. We observe that the error gen-erally decreases with increasing tag frequency. There are tworeasons for this. First, there is less overlap with the dc spec-tral peak as frequency increases, reducing the associated arti-facts. Second, the larger number of pixels within the larger band-pass region at the higher frequencies gives more informationabout the modulated motion spectrum. Finally, the reason thatthe error increases at tag period 2 mm is that high-frequency

parts of the two spectral peaks were truncated in the Fourier do-main, i.e., the highest frequencies were not imaged. To studythe effect of image noise, we added different levels of whiteGaussian noise to the simulated images and computed the rmsdisplacement errors. These results are summarized in Fig. 5(b),where the rms displacement error is plotted against the noiseratio for four tag periods. The noise ratio is defined as the ratioof the noise standard deviation to the maximum image magni-tude (the reciprocal of the SNR is used so that we can plot thezero-noise case). As is typical in FM and PM communicationssystems, we see that the performance degrades gradually as thenoise increases.

B. Magnitude and HARP Images

1) Magnitude Images:So far, we have concentrated entirelyon the angle of the harmonic imagebecause it contains infor-mation about cardiac motion. The magnitude (modulus) ofis also useful, however, because it contains information aboutcardiac geometry. Consider the horizontal and vertical planartagged image sequences shown in Fig. 6. These images have64 64 pixels with 1.25-mm pixel separation, cropped from theoriginal images to show close-up short-axis images of the de-forming LV during systole. The first image in each sequencecorresponds to 47.3 ms after end diastole and subsequent imagesare 32.5-ms apart. Using the spectral peaks corresponding to thefirst harmonic of each image, we used the optimal bandpass fil-ters to compute two sequences of harmonic images. Then, foreach time frame, we computed the average of the magnitudesof the two harmonic images produced from the two tag orien-tation images. An image sequence showing the average magni-tude image is shown in Fig. 7(a). The quality of these images isquite low because we are using only a very small part of Fourierspace to reconstruct the geometry.


(a)

(b)

Fig. 5. (a) Optimal radii of the elliptical bandpass filter as a function of tag period: rms displacement error as a function of tag period. (b) RMS displacementerror versus noise fraction for different tag periods.

As tags fade, the energy of the signal in the neighborhood of aspectral peaks diminishes. Since there is no dc component in abandpass filter, the reduction in intensity evident in the imagesequence of Fig. 7(a) reflects this tag fading. The calculatedenergy in a harmonic image can, in fact, be used to compen-sate for the effects of tag fading by simply multiplying eachimage so that it has the same energy as the first image. Ap-plying this simple correction scheme to the images in Fig. 7(a)yields the corresponding normalized images in Fig. 7(b). Al-though far from ideal representations of the myocardial geom-etry, these images can be used in fast segmentation for visual-ization of computed motion quantities within the myocardium.

2) Examples of HARP Images:HARP images computedusing the first harmonic of the images in Fig. 6 are shown inFig. 8. For display purposes, they are shown overlayed on a

simple threshold segmentation of the normalized magnitudeimages in Fig. 7(b). While it is somewhat misleading to lookat the very jagged-looking HARP images themselves, a certainkey point can be made. We observe that the bending of thesaw-tooth pattern in the HARP images is similar to the bendingof the tag lines in the original images. This reflects the fact thatthe HARP angle is a material property of the tissue that followsthe motion of the LV.

C. Motion Estimation Experiments

1) Synthetic Tag Lines:Fig. 9 shows three examples of syn-thetic tag generation. In all three cases, the synthetic tags werecreated using an isocontour algorithm, and the resulting lineswere manually trimmed to remove lines outside of the region


Fig. 6. A planar tagged image sequence with (a) horizontal and (b) vertical tags showing the short-axis of the LV in systole.

of interest. A very important observation in Fig. 9 is the agree-ment between the synthesized tag lines and the original tags ofthe image, giving strong evidence that HARP images contain thesame motion information as the original images. In Fig. 9(a), asingle HARP image was generated from the spectral peak cor-responding to the first harmonic of the underlying image. Thenthe HARP angle isocontour valuewas set to so that the gen-erated lines coincided with the tag lines in the image. The linesgenerated in this fashion appear to very accurately track the taglines in the image. In Fig. 9(b), two HARP images were used,one at 45 and the other at 135. For each HARP image, the iso-contour angle was manually adjusted until the generated linesappeared to coincide with the visible tag lines within the image.

The final example, shown in Fig. 9(c), demonstrates how twoisocontour values can be used to generate synthetic tag lines thatare closer together than the tag lines appearing in the image andthat these lines do not have to coincide with those in the image.This particular image data is from a canine heart abnormallyactivated by a pacing lead placed at the base of the free LV wall

(approximately the one o'clock position) [48]. In this relativelyearly systolic image, early mechanical activation near the pacinglead is seen as the tag lines bending inward toward the LV cavity.Associated prestretching is seen in the septal wall, where the taglines bend outward toward the right ventricle (RV) cavity. Thisis the correct pattern, which demonstrates that HARP imagescan be applied to abnormal cardiac motion as well as normalmotion.

2) Measuring Small Motion:Using (18), we computed theapparent 2-D displacement field given the horizontal andvertical tagged images at the second time frame in the imagesequences (shown in Fig. 6). The resulting computed displace-ment field, scaled and decimated by a factor of two, is shownin Fig. 10. Here, the magnitude image was also used to createa simple threshold segmentation that masks the display of dis-placement vectors to those nominally in the myocardium.

Several observations can be made from Fig. 10. First, theoverall motion pattern appears to be very smooth or coherent.


Fig. 7. (a) Average magnitude of the complex images computed from the tagged images in Fig. 6. (b) These same images normalized to account for tag fading.

We emphasize that this is not the result of regularization or spa-tial smoothing. Our results are computed on a pixel-by-pixelbasis. In fact, vectors outside the myocardium (not shown) arequite chaotic looking. Second, the overall pattern is one of coun-terclockwise rotation and careful scrutiny reveals that there isa more inward orientation of the vectors nearer the LV cavitythan those farther away. This is evidence of both contractionand thickening, which we know must occur in early systole.Third, there are fairly coherent vectors appearing on the pap-illary muscles that extend into the LV cavity. This is initial evi-dence, though not conclusive by any means, that we may be ableto track these muscles as well as those within the larger myocar-dial wall.

3) Measuring Strain:Next, we used the HARP images inFig. 8 to compute the apparent 2-D circumferential strain duringLV contraction. In order to do this, we manually specified acenter point for the LV and computed unit vectors that aretangents to circles centered at this point. Equation (24) was then

used to compute , which represents the apparent 2-D circum-ferential strain in this case. The quantity displayed in Fig. 11 is

smoothed by a 7 7 averaging filter and then restrictedto a simple threshold segmentation obtained from the magnitudeimages. The field is smoothed to compensate for noise in thegradient computation than the simple averaging filter we em-ployed in this paper.

There are several observations we can make from Fig. 11.First, we note that this image is a midventricle LV short-axisimage of a normal subject. The midventricle is typically un-dergoing a very simple out-of-plane motion, basically a trans-lation and compression of the base toward the apex. There-fore, the apparent 2-D strain in a short-axis plane is a veryuseful diagnostic quantity. Second, overall darkening of the my-ocardial strain map indicates that its circumference is gettingshorter, i.e., the LV is contracting. Third, there are lighter spotsnear the six and ten o'clock positions that indicate less short-ening than in other regions. These locations happen to be thelocations where the right ventricular myocardium joins the LV


Fig. 8. (a) Horizontal and (b) vertical angle images derived from the tagged images in Figs. 6(a) and (b), respectively.

and this behavior is expected. Finally, a slightly lighter shadethroughout the septum is expected as this region normally doesnot undergo quite as much contraction as the free wall [8].

To demonstrate how our strain estimation procedure performson abnormal cardiac motion, we computed the apparent 2-D cir-cumferential strain on a full set of planar tagged image data froman abnormally paced canine heart [48]. A single image fromthis data set was shown in Fig. 9(c). A time sequence of com-puted 2-D circumferential strain estimates is shown in Fig. 12,restricted to the automatically generated myocardial segmenta-tion obtained from the magnitude images. The time interval be-tween the images is 19.5 ms and the tag period is 5.5 mm (fora complete description of the experiment see [48]). The thirdimage in this sequence reveals early activation (circumferen-tial contraction) in the one o'clock position, the location of thepacing lead, which persists for several frames. The lighter re-gion in the septum also visible in the third frame indicates aprestretch of the myocardial fibers which persists well into thesystolic phase. This indicates a significant delay of the onset of

shortening because the conduction does not travel through thenormal pathways of the heart. Many of the basic conclusionspresented in the 3-D analysis in [48] are evident in our resultsobtained in a fast completely automatic fashion.

V. DISCUSSION ANDCONCLUSION

Perhaps the most important aspect of the method presentedhere is its computational simplicity in producing measures ofmotion. The basic mathematical operations here include: the in-verse Fourier transform to produce a complex image; computingthe HARP images; and computing the derivatives of the HARPimages. These computations are fast and in few minutes the re-sulting strain maps are produced. For example, for the canineheart with 20 time frames, it took only three min on a 350-MHzPentium II computer to compute the strain. Bearing in mind thatthe code was implemented using MATLAB, there is clearly po-tential for even faster computations. Also important, is the factthat the only human intervention required was the setting of the


Fig. 9. Synthesized tag lines generated from (a) the first harmonic peak of this higher-order SPAMM-tagged image, (b) two diagonal angle images of this 2-D1-1 SPAMM-tagged image, and (c) from the first harmonic peak this higher order SPAMM-tagged image of a canine heart.

threshold level for the rough segmentation and picking the longaxis position in the images to determine the radial and circum-ferential directions. The other settings, such as the filters sizeand location, were determined automatically from the taggingseparation and orientation, which was set by the scanner’s op-erator.

In the experiments presented in this paper, we have computedHARP images by extracting the spectral peaks from existingtagged MR images. Since MRI is a Fourier imaging technique,it would be more efficient to acquire only the Fourier data oneneeds directly from the scanner. This idea has promise for fastmotion imaging using MR tagging. One approach would be touse separate 1-D SPAMM acquisitions, imaging their respec-

tive spectral peaks in separate phase-encoded sets of acquisi-tions. Another approach would be to use 2-D SPAMM acqui-sitions, orienting the read-out direction to acquire two (orthog-onal) spectral peaks in one phase-encoded sets of acquisition.The increased speed of acquisition could be used to shortenbreath-holds, to acquire images closer together in time, or to ac-quire denser 3-D image sets. On the other hand, the acquisitionof separate spectral peaks might be susceptible to field inhomo-geneity, so further research has to be done.

Interference from spectral peaks is a source of artifacts inour method and the dc peak is particularly problematic since itsenergy increases as the tags fade. Using higher frequency tagshelps but, because of frequency truncation and the requirement


Fig. 10. The displacement field calculated from the second pair of tagged images in the image sequence of Fig. 6.

to image larger regions in Fourier space, there is an upper boundon usable tag frequency. It turns out that direct Fourier acquisi-tion from the scanner can help this problem as well. In particular,the use of 180 total tag tip angle will largely suppress the dccomponent in the early stages of acquisition. The deleterious ef-fects of tag fading will be delayed as well because of the largertip angle. To image later parts of the cardiac cycle (diastole),however, it would still be necessary to use midcycle tagging.

Interference from adjacent spectral peaks, including thedc peak and higher frequency peaks, can produce artifactsin the HARP methods. One example of an artifact is theblotchy or pinwheel effect seen in HARP strain maps. Anotherexample is the branching and truncations of the HARP imageswithin the myocardium, which occurs on rare occasions. DirectFourier acquisition with 180 tip angles should help thisproblem somewhat because it suppresses the dc lobe. Theuse of 1-1 SPAMM should help because there will be nohigher order spectral peaks with which to interfere. Also,

in 2-D 1-1 SPAMM, the 180 total tip-angle protocol alsosuppresses the cross terms, leaving essentially a four-peakpattern. Furthermore, the cross term peaks do not reappearas the tags fade, as does the dc peak. This bodes well forthe very fast acquisition of two peaks at once, using 2-D 1-1SPAMM. Further suppression of noise in strain computationmay require a more sophisticated regularization approaches.

Because our emphasis has been on reducing image acquisi-tion times, our approach describes the use of just two spectralpeaks in Fourier space. The data we actually used in our experi-ments, however, had more peaks available in the data and thesecould have been exploited as well. For example, using the eightspectral peaks (not counting the dc peaks) available in the studyof Fig. 6 might have led to improved results. Since acquiringmore data in the read-out direction has virtually no extra costassociated with it, it may be beneficial to extend our methods toinclude linearly dependent vectors. This would require least


Fig. 11. The estimated circumferential strain from the complete set of the planar tagged images of the human LV shown in Fig. 6. The gray level color barindicates the strain value, varying from the darkest level with 25% shortening to the lightest with 10% stretching.

Fig. 12. The circumferential strain in a short-axis image of a paced canine LV. The gray level color bar indicates the strain value, varying from the darkest levelwith 20% shortening to the lightest with 20% stretching.

squares methods and would presumably lead to more accurateresults (cf. [32]).

There are several ways to extend our results in this paper to3-D. One approach is to use our synthetic tag lines in any cur-rently existing tag line analysis methods based on orthogonal

image acquisitions [15]–[17]. There should be an immediateadvantage coming from the denser tag lines we can generate,although the approach would still be limited by the density ofimage planes. Another approach is to consider tag directionsnot oriented in the image plane [40]. In this case, (11) becomes


the starting point for extending our methods to 3-D. It turns outthat it is possible to reconstruct 3-D displacement vectors andthe full 3-D strain tensor given adequate 3-D image data. Weare actively exploring this in our current research program, andit will be the subject of a future paper.

APPENDIX

A. Proof of Small Displacement Calculations

Since the phase differs from the HARP angle by a mul-tiple of , it follows that

This can be further simplified using (8) and (11), leading to

Now if , then no wrapping will occur and

if (30)

From (30) we get and (18) followsby simple rearrangement.

B. Proof of Strain Calculations

Since the phases and are the same for and , we canuse (11) and the fact that to show that

for . Defining and rearranging yields

(31)

which can be solved for , yielding

(32)

Taking the gradient of (32) yields (20).

ACKNOWLEDGMENT

The authors are grateful to M. Guttman, B. Kerwin, and S.Gupta for their comments and suggestions.

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